A revisit of the Gram-Charlier and Edgeworth series expansions

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In this paper we make several observations on the Gram-
Charlier and Edgeworth series, which are methods for modeling
and approximating probability density functions.We
present a simplified derivation which highlights both the
similarity and the differences of the series expansions, that
are often obscured by alternative derivations. We also introduce
a reformulation of the Edgeworth series in terms
of the complete exponential Bell polynomials, which make
both series easy to implement and evaluate. The result is
a significantly more accessible methodology, in the sense
that it is easier to understand and to implement. Finally,
we also make a remark on the Gram-Charlier series with a
gamma kernel, providing a novel and simple expression for
its coefficients.